Perhaps your CPU has died, yes. Or, hopefully, it could be a dead CPU and/or case fan that is causing your comp to shut down via some anti-overheating BIOS function. Unlikely, but possible.
You say you swapped out the PSU? Try a third if you can, and make sure all the power cables are properly inserted.
Check your graphics card too, could have died/is dieing and is stopping your system from POSTing.
AMD X2 4400+ Toledo 1MB/Asus A8N-SLI Deluxe/2GB Kingston HyperX/SLI Gigabyte 7800 GTX 256MB/Creative SB Audigy 2 ZS Plat w/ Creative Inspire 6.1 speakers /180GB Hitachi PATA/19\" AG Neovo F-419 LCD/OCZ PowerStream 600W PSU/WinXP Home SP2